A complete normed lattice is called a Banach lattice. An operator T : E ! X from a Banach lattice E into a Banach space X is called a KB-operator if for every positive increasing sequence (xn) in the closed unit ball of E, (Txn) converges in X. An operator T : E ! E on a Banach lattice E is called unbounded demi KB-operator if for every positive increasing sequence (xn) in the closed unit ball of E such that (xn - Txn) is unbounded norm convergent to x 2 E , there is an unbounded norm convergent subsequence of (xn). In this presentation, we study the unbounded demi KB-operators on a Banach lattice.